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Theorem tendoplcbv 36450
Description: Define sum operation for trace-perserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)
Hypothesis
Ref Expression
tendoplcbv.p 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
Assertion
Ref Expression
tendoplcbv 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
Distinct variable groups:   𝑡,𝑠,𝑢,𝑣,𝐸   𝑓,𝑔,𝑠,𝑡,𝑢,𝑣,𝑇
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑓,𝑔,𝑠)   𝐸(𝑓,𝑔)

Proof of Theorem tendoplcbv
StepHypRef Expression
1 tendoplcbv.p . 2 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
2 fveq1 6271 . . . . 5 (𝑠 = 𝑢 → (𝑠𝑓) = (𝑢𝑓))
32coeq1d 5359 . . . 4 (𝑠 = 𝑢 → ((𝑠𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑡𝑓)))
43mpteq2dv 4821 . . 3 (𝑠 = 𝑢 → (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))))
5 fveq1 6271 . . . . . 6 (𝑡 = 𝑣 → (𝑡𝑓) = (𝑣𝑓))
65coeq2d 5360 . . . . 5 (𝑡 = 𝑣 → ((𝑢𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑣𝑓)))
76mpteq2dv 4821 . . . 4 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))))
8 fveq2 6272 . . . . . 6 (𝑓 = 𝑔 → (𝑢𝑓) = (𝑢𝑔))
9 fveq2 6272 . . . . . 6 (𝑓 = 𝑔 → (𝑣𝑓) = (𝑣𝑔))
108, 9coeq12d 5362 . . . . 5 (𝑓 = 𝑔 → ((𝑢𝑓) ∘ (𝑣𝑓)) = ((𝑢𝑔) ∘ (𝑣𝑔)))
1110cbvmptv 4826 . . . 4 (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔)))
127, 11syl6eq 2742 . . 3 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
134, 12cbvmpt2v 6820 . 2 (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))) = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
141, 13eqtri 2714 1 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1564  cmpt 4805  ccom 5190  cfv 5969  cmpt2 6735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pr 4979
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-br 4729  df-opab 4789  df-mpt 4806  df-co 5195  df-iota 5932  df-fv 5977  df-oprab 6737  df-mpt2 6738
This theorem is referenced by:  tendopl  36451
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